Problem 22 - Arihant AIEEE Physics

Question

\(n\) transparent slabs of refractive index \(1.5\) each having thickness \(1 \mathrm{~cm}, 2 \mathrm{~cm}, \ldots\) to \(n \mathrm{~cm}\) are arranged one over another. A point object is seen through this combination with near perpendicular light. If the shift of object by the combination is \(1 \mathrm{~cm}\) then the value of \(n\) is: (a) either 2 or 3 (b) 2 (c) 3 (d) \(0.3\)

Step-by-Step Solution

Answer

The value of \( n \) is either 2 or 3 (option a).

Step 1: Understanding the problem

The problem involves finding the number of transparent slabs, each with increasing thickness, that result in a total apparent shift of 1 cm when viewed through the slabs.

Step 2: Defining the formula for apparent shift

The apparent shift (or displacement) in each slab can be computed using the formula: \( D = t(1 - \frac{1}{n}) \), where \( t \) is the thickness of the slab and \( n \) is the refractive index. Here, \( n = 1.5 \).

Step 3: Applying the formula for each slab

Compute the individual shift for each slab: the first slab has thickness \( 1\, \text{cm} \), the second \( 2\, \text{cm} \), continuing up to the \( n \)-th slab with thickness \( n\, \text{cm} \). Compute these using \( D_i = t_i (1 - \frac{1}{1.5}) \).

Step 4: Calculate the total shift

Sum up the shifts for all slabs: \( D_{total} = \sum_{i=1}^{n} t_i (1 - \frac{1}{1.5}) \). The formula \( 1 - \frac{1}{1.5} = 0.3333 \) simplifies the shift calculation for each layer to \( D_i = 0.3333 t_i \).

Step 5: Setting the total shift to 1 cm

Set the total shift we calculated equal to the given total shift of 1 cm: \[ 0.3333 \sum_{i=1}^{n} i = 1 \].

Step 6: Solve for n

Recognize that \( \sum_{i=1}^{n} i = \frac{n(n+1)}{2}\) (the sum of the first \( n \) natural numbers). Set \[ 0.3333 \frac{n(n+1)}{2} = 1 \] to find \( n \).

Step 7: Calculate \( n \)

Simplify the equation: \[ \frac{n(n+1)}{6} = 1 \], which gives \( n(n+1) = 6 \). Solving for \( n \), possible integer solutions are \( n = 2 \) or \( n = 3 \).

Step 8: Final result

We found that the calculated shifts can match the condition for \( n = 2 \) or \( n = 3 \). Therefore, choice (a) is correct.

Key Concepts

Apparent Shift

When an object is observed through a series of transparent materials, like glass or plastic slabs, it often seems to be at a different position than its actual location. This is what we call an **apparent shift**.
It occurs because light rays bend as they pass from one medium to another, in a process called refraction. The apparent shift depends on two main factors:

• **Thickness of the slab**: The thicker the slab, the greater the potential shift.
• **Refractive index of the material**: This defines how much light bends when entering the slab.

The formula used to calculate the shift is: \[D = t \left( 1 - \frac{1}{n} \right)\]where \( D \) is the apparent shift, \( t \) is the thickness, and \( n \) is the refractive index. By understanding these aspects, students can predict how an object might appear from another position when looking through transparent materials.

Refractive Index

The **refractive index** is a crucial concept in geometric optics. It measures how much a medium can bend light. When light enters a medium with a refractive index different from air (which is approximately 1), it changes speed, bending at the interface.

• **Higher Refractive Index**: This means the medium is more effective at bending light, leading to a smaller critical angle and greater shift.
• **Common Materials**: Glass often has a refractive index around 1.5, while water is about 1.33.

A high refractive index results in a more significant apparent shift, affecting how we perceive objects behind the medium.

Transparent Slabs

Transparent slabs, such as layers of glass stacked together, have practical applications and intriguing optical properties that challenge the way we see. These involve:

• **Stacking Effects**: Multiple slabs compound the apparent shift effect. Each layer adds to the total shift, calculated as a sum of shifts from individual slabs.
• **Practical Uses**: In magnifying glasses and camera lenses, understanding the behavior of transparent slabs helps in engineering precise optical devices.

In stacks, the light experiences multiple refractions, each adding to the total shift according to the formula for apparent shift. This phenomenon illustrates how the alignment and thickness of slabs can fine-tune how light paths and images are perceived.